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To do so, we show that these are precisely the groups that admit a presentation of the form $ \\label{tau2pres_0}\\langle A, C \\mid [a_i, a_j]= \\prod_t {\\scriptstyle c_t^{\\scriptscriptstyle \\lambda_{t,i,j}}} \\ (i< j), \\ [A,C]=[C,C]=1\\rangle,$ where $A=\\{a_1, \\dots, a_n\\}$, and $C=\\{c_1, \\dots, c_m\\}$. Hence, one may select a random $\\tau_2$-group $G$ by fixing $A$ and $C$, and then randomly choosing exponents $\\lambda_{t,i,j}$ with $|\\lambda_{t,i,j}|\\leq \\ell$, for some $\\ell$.\n  We prove th"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.02651","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-08T14:11:44Z","cross_cats_sorted":[],"title_canon_sha256":"396610571a334bd45f02400dfc676c1d6a2a22b6692c7792b544888c2dd05179","abstract_canon_sha256":"70153bc1a9653cf0a6afb8963f656483480c919fc049a07474a1a04615d7cd62"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:33.170155Z","signature_b64":"NVZ9/YI6U1FBYVuYh5NcqIGIBlSmrU51HnjElJtOSE8o5cRSN+Q8xoB2lrBsCFxuWjtzHWQmuRtLSVE6P4u1Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ed6d955ea02346de309eb710f1737b41498949a7b015246b44163b357316cb5","last_reissued_at":"2026-05-18T00:55:33.169629Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:33.169629Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random nilpotent groups, polycyclic presentations, and Diophantine problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov","submitted_at":"2016-12-08T14:11:44Z","abstract_excerpt":"We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $\\tau_2$-groups). To do so, we show that these are precisely the groups that admit a presentation of the form $ \\label{tau2pres_0}\\langle A, C \\mid [a_i, a_j]= \\prod_t {\\scriptstyle c_t^{\\scriptscriptstyle \\lambda_{t,i,j}}} \\ (i< j), \\ [A,C]=[C,C]=1\\rangle,$ where $A=\\{a_1, \\dots, a_n\\}$, and $C=\\{c_1, \\dots, c_m\\}$. Hence, one may select a random $\\tau_2$-group $G$ by fixing $A$ and $C$, and then randomly choosing exponents $\\lambda_{t,i,j}$ with $|\\lambda_{t,i,j}|\\leq \\ell$, for some $\\ell$.\n  We prove th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02651","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.02651","created_at":"2026-05-18T00:55:33.169718+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.02651v1","created_at":"2026-05-18T00:55:33.169718+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02651","created_at":"2026-05-18T00:55:33.169718+00:00"},{"alias_kind":"pith_short_12","alias_value":"D3LNSVPKAI2G","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"D3LNSVPKAI2G3YYJ","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"D3LNSVPK","created_at":"2026-05-18T12:30:09.641336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ","json":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ.json","graph_json":"https://pith.science/api/pith-number/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/graph.json","events_json":"https://pith.science/api/pith-number/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/events.json","paper":"https://pith.science/paper/D3LNSVPK"},"agent_actions":{"view_html":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ","download_json":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ.json","view_paper":"https://pith.science/paper/D3LNSVPK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.02651&json=true","fetch_graph":"https://pith.science/api/pith-number/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/graph.json","fetch_events":"https://pith.science/api/pith-number/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/action/storage_attestation","attest_author":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/action/author_attestation","sign_citation":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/action/citation_signature","submit_replication":"https://pith.science/pith/D3LNSVPKAI2G3YYJ5NYQ6FZXWQ/action/replication_record"}},"created_at":"2026-05-18T00:55:33.169718+00:00","updated_at":"2026-05-18T00:55:33.169718+00:00"}